神部, 勉

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Titleオイラーの方程式導出と流体運動のゲージ理論 (オイラー方程式250年 : 連続体力学におけるオイラーの遺産)Author(s)神部, 勉Citation数理解析研究所講究録 (2008), 1608: 14-25Issue Date2008-07URL http://hdl.handle.net/2433/139996RightType Departmental Bulletin PaperTextversionpublisherKyoto University( )(Tsutomu Kambe)Chern Institute ofMathematice(Nankai University), , (1)?(2). ?(3).. PartI( ). Part II. LEOO$=$ Leonhard Euler, Opera Omnia( ).Part I:1.18‘ ’ ‘ ’25020 [1,21“3 : eonhard Eu-ler (Meda on execute par Dominique Rachette, 1781). [LEOO, Ser.II, vol.12 ]3 $(x,y,z)$ $m$ $(X, Y,Z)$$t$ $(x(t),y(t),z(t))$1608 2008 14-25 14$m \frac{d^{2_{X}}}{dt^{2}}=X(x,y,z)$ , $m \frac{d^{2}y}{dt^{2}}=Y(x,y,z)$ , $m \frac{d^{2}z}{dt^{2}}=Z(x,y,z)$ (A)$t$ 2(1687)II “. ”() ?II[31 ()$\omega$[4].1960 [1]1750 [6]1736 [5] , ‘ ’ ‘( ) ’$[3]_{0}$$\tau$2.1736 [5131747 [4]. 20‘ ‘‘ ‘ $m$( ), 1750[6] :$m$ d& $=$ P $dt^{2}$ , $mddy=Qdt^{2}$ , $m$ d& $=$ Rdt2 (B)( $m$ $2M$ [4])$\omega$15183(B)(B) [1, 6].(I) (B)(II) 2(III)(B)(IIl)$[$3, $4]$ “” [2, 81.(B) 1 $(X, YZ)$1 ‘’3.$[8]_{0}$(corpuscles)100( )general principle$[$2, $8]_{0}$““( corpuscles)$a^{2}$ $pa^{2}$ $p$$\rho$(a)3$(u_{J}V_{J}w)$ $V$16$q$ $V$ $qV$ $dt$ $V$ $V^{\dagger}$$q$ $q$ ’ $q’ V$‘ ( ) ’[2, 6, 8] :$( \frac{dq}{dt})+(\frac{dqu}{\ }1+( \frac{dqv}{\phi}1+(\frac{dqw}{\ } I=0$ ($C$)(dq/dt) 4 / $t$$()$$q$$( \frac{du}{\ })+(\frac{dv}{\phi})+(\frac{dw}{\ } I=0$ (D)(b)1755 [9] ( 1757)$V$ $m=pV$$F=(PQ, R)$$G$$m\vec{A}=\vec{G}+m\overline{F}$ (E)($A$ )$x$ Ar:$A_{X}=( \frac{du}{dt}I+u(\frac{du}{\ })+v( \frac{du}{\phi})+w(\frac{du}{\ })$ . (F)$x,$ $J^{r}$ 2 3 $\Delta x,$ $\Delta y,$ $\Delta$ 2$x$ $x$ (i2$G_{X}=- \Delta \mathfrak{r}\Delta y\ ( \frac{dp}{\ })=-\frac{m}{q}(\frac{dp}{\ } I$ $(\Delta x\Delta y\Delta z=m/q)$$x$ ,$mA_{x}=-(m/qXdp/\ )+mP$ (G). (F) $m$17$( \frac{du}{dt})+u(\frac{du}{dx})+v(\frac{du}{dy})+?l)(\frac{du}{dz})=-\frac{1}{q}(\frac{dp}{dx})+P$ ( $H$ )(B), (C), (D), (H)$()$(C)$K$$m\ddot{r}=F^{(ext)}$ ,$0=F^{(exl)}-m\ddot{r}$ ,$K’$$K’$ $m\ddot{r}$2(G) ()(G)250Part II: Gauge principleEulerirrotational$[$ 10, 11, $12|$4 $a$-4.1$a=(a^{1}, a^{2}, a^{3})=(a, b, c)$$a^{\mu}=(\tau, a^{1}, a^{2}, a^{3})$$\mu$ 0,1,2,3 , $a^{0}$$\tau(=t)$ ( )$x=(x, y, z)=(x1x^{2}, x^{3})$ $t$ $a$ $X^{k}(a^{\mu})$ ,$(X^{k})=(X, Y, Z)$ $v^{k}=\partial_{\tau}X^{k}$ , $X_{0}^{k}$18, $d^{3}a=dadbdc$ $d^{3}x=dxdydz$ $dm$. $dm=d^{3}a=\rho d^{3_{X}}$ ( $\rho$ ).$\partial_{\tau}(dm)\equiv’\partial_{\tau}(d^{3}a)=0$ . (1). $dm=dadbdc=\rho dXdYdZ$$\rho=\frac{1}{J}$ . $J= \frac{()(X^{1}X^{2},X^{3})}{\partial(a^{1}’,a^{2},a^{3})}=\frac{(\prime(X,Y,Z)}{\partial(a)b,c)}$ . (2)$J$ a-x Jacobian$s$ $s=s(a, b, c)$$\partial_{\tau^{S}}=0$ . (3)$\epsilon,$$h$$\delta\rho$ , $\delta$$\delta\epsilon=(\delta\epsilon)_{s}=\frac{p}{\rho^{2}}\delta\rho$ , $( \frac{\partial\epsilon}{\partial\rho})_{s}=\frac{p}{\rho^{2}}$ , $\delta h=\frac{1}{\rho}\delta p$ , (4)$p$ $h=\epsilon+p/\rho$ $(\cdot)_{s}$ $s$Lagrange $L$ , $I$ $(k, l=1,2,3)$ :$AT$ $= \int_{I_{a}}\frac{1}{2}X_{\tau}^{k}X_{\tau}^{k}d^{3}a-/\Lambda i_{a}\epsilon(\rho, s)d^{3}a$ , (5)$I$ $=$ $/L(X_{l^{J}}^{k})d^{4}a$ , $d^{4}a=d\tau d^{3}a$ , (6)$L(X_{\mu}^{k})= \frac{1}{2}X_{0}^{k}X_{0}^{k}-\epsilon(X_{l}^{k}, a^{k})$. (7)$\epsilon(X_{l}^{k}, a^{k})$ $\epsilon(\rho, s)$$\rho$ $X_{l}^{k}$ $s$$\grave\grave$ $a^{k}$42(7) $L$ .$\frac{\partial}{\partial’a^{\mu}}(\frac{\partial L}{\partial X_{\mu}^{k}})-\frac{\partial L}{\partial^{r}X^{k}}=0$ , $X_{\mu}^{k}= \frac{\partial X^{k}}{\partial a^{\mu}}$ (8)$T_{\mu}^{\nu}$ [12] :$T_{\mu}^{\nu} \equiv X_{\mu}^{k}(\frac{\partial L}{\partial X_{\nu}^{k}})-L\delta_{\mu}^{\nu}$ , (9)$k=1,2,3$. :$\frac{\partial}{\partial a^{\nu}}T_{\mu}^{\nu}=0$ . (10)19$(a)\mu\neq 0(x^{\mu}=\alpha)$ $\partial_{\nu}T_{\mu}^{\nu}=0$ :$\partial_{\tau}V_{\alpha}+\partial_{\alpha}F=0$ $(V_{\alpha}\equiv X_{a}X_{\tau}+Y_{\alpha}Y_{\tau}+Z_{\alpha}Z_{\tau})$ , (11)$F=- \frac{1}{2}v^{2}+h$ . 2 $\alpha$ $(a, b, c)$ $\tau$$0$ $t$ Wcber [13, \S 15]:$V_{\alpha}(\tau)$ $\equiv$ $X_{lX}X_{\tau}+Y_{tX}Y_{\tau}+Z_{\alpha}Z_{\tau}=V_{\alpha}(0)-d_{\alpha}\chi$ , (12)$\chi=/o^{t}Fd\tau=/o^{t}(-\frac{1}{2}v^{2}+h)d\tau$ . (13)(11) $V_{\alpha}$ $a$- (5.1 ). $V_{\alpha}(O, a)$ $h(O, a)(a=x)$(12), (13)$(b)\mu=0$ , :$(\prime J_{\tau}H$ $+$ $\partial_{a}[p\frac{\partial(X,Y,Z)}{(\prime’(\tau_{\tau}b,c,)}]+\partial_{b}[p\frac{\partial(X,Y,7\lrcorner)}{(’(a,\tau_{!}c_{d})}]$$+$ $\partial_{(\vee}.[p\frac{\partial(X,Y,Z)}{\partial(a1b)\tau)}]=0$ . (14)$H= \frac{1}{2}v^{2}+\epsilon$ . (11) $(\tau, a)$ :$A_{a} \equiv X_{\alpha}X_{\tau\tau}+Y_{\alpha}Y_{\tau\tau}+Z_{\alpha}Z_{\tau\tau}=-\frac{1}{\rho}\partial_{\alpha}p$. (15)[13, \S 13].:$X_{\tau\tau}=- \frac{1}{\rho}\partial_{x}p$, $\partial_{x}p=\frac{\partial\alpha}{(r)_{X}}\frac{\partial p}{\partial\alpha}$ . (16)$X_{\tau\tau}$ $a$ $P$ (G)(3 (b)) $L$ ( )43 ( )(15) $a$- $x$-2 $x$- $(X_{\tau\tau}, Y_{\tau\tau}, Z_{\tau\tau})$$a$- $\alpha$- $(X_{\alpha}, Y_{\alpha}, Z_{\alpha})$(15) $x$- $\Delta X=(\Delta X, \Delta Y, \Delta Z)$(12) $V_{\alpha}(\tau, a)$5 $x$-51$(t, x, y, z)$ . [11,$12|$ . $\partial_{\tau}$ $D_{t}$ :$\partial_{\tau}=D_{t}$ , $D_{t}\equiv\partial_{t}+\prime v^{k\prime}\partial_{k}=\partial_{t}+u\partial_{x}+v\partial_{y}+w\partial_{z}=\partial_{t}+v\cdot\nabla$ . (17)20$D_{t}$ $v(x, t)$ :$v(x, t)=\partial_{\tau}^{t}X=D_{t}x$ . (18)$\mathcal{A}(x, t)$ :$\mathcal{A}(x, t)=\partial_{\tau}^{2}X=D_{t}v=(\partial_{t}+v^{k}\partial_{k})v$. (19)41 $d^{3}a(a)$ $s=s(a)$ (1)(3) . 2 :$L_{\phi}=-/\Lambda i\partial_{\tau}\phi d^{3}a$ , $L_{\psi}=-/ \rfloor\iota’\int s’\partial_{\tau}\psi d^{3}a$. (20)$\phi(a, \tau)$ $\psi(a, \tau)$ (potentiaJs)(5) AT $L_{\phi}$ $L_{\psi}$ Lagrangian $A_{T^{*}}$ :$\Lambda_{T}^{*}=\Lambda_{T}-/\partial_{\tau}\phi d^{3}a-/s’\partial_{\tau}\psi d^{3}a$ . (21),$I=/\tau 1T_{2}\Lambda_{T^{*}}d\tau=/d\tau\Lambda_{T}-/d\tau/\partial_{\tau}\phi d^{3}a-/d\tau/s\partial_{\tau}\psi d^{3}a$. 2 $I_{\phi}= \int d\tau\int\partial_{\tau}\phi d^{3}a$ $\tau$ $\int[\phi]d^{3}a$. $[\phi]=\phi|_{\tau}2-\phi|_{\tau_{1}}$ 2 $\tau_{2},$ $\tau_{1}$ $\phi$ $\tau\in(\tau_{1}, \tau_{2})$ $\phi$3 $s$ $\tau$ $I_{\psi}= \int[\psi]sd^{3}a$ .$\tau\in(\tau_{1}, \tau_{2})$ $I$ ( ) $\phi,$ $\psi$$x$- nontrivial. $d^{3}a=\rho d^{3_{X}}$ $\partial_{\tau}=D_{t}$ $L_{\phi}=$$- \int_{M}\rho D_{l}\phi d^{3_{X}}$, $L\psi=-._{M}\rho sD_{t}\psi d^{3_{X}}$$x$- Lagra gian $\Lambda_{T^{*}}=\int_{AI}\mathcal{L}(v, \rho, 1i’, \phi, \psi)d^{3}x$ $\mathcal{L}$:$\mathcal{L}\equiv\frac{1}{2}\rho v^{k}v^{k}-\rho\epsilon(\rho, s)-\rho D_{t}\phi-\rho sD_{t}\psi$ (22)x- $I= \int \mathcal{L}(v, \rho, s, \phi, \psi)d^{4_{X}}$$(d^{4}x=dtd^{3_{X}})$ .$s_{0}$ ( $[11|)$ .52$I$ : $x’(x, t)=$$x+\xi(x, t)$ . $d^{3_{X}}$ 1 $d^{3}x’=(1+\partial_{k}\xi^{k})d^{3_{X}}$$\Delta(d^{3}x)=\partial_{k}\xi^{k}d^{3_{X}}$$\Delta\rho=-\rho\partial_{k}\xi^{k},$ $\Delta v=D_{t}\xi$ , $\Delta s=0$ (1) (3) $(\phi,$ $\psi$), $I$ :$\Delta I=/d^{4_{X}}[\frac{\dot{\mathfrak{c}}JL}{\partial v}\Delta v+\frac{\partial’L}{\partial\rho}\Delta\rho+\frac{\partial L}{\partial s}\Delta s+L\partial_{k}\xi^{k}]$ .21$\xi^{k}$ .$\frac{\partial}{\partial t}(\frac{\partial L}{\partial_{t^{1k}}})+\frac{\partial}{\partial x^{l}}(v^{l}\frac{\partial L}{\partial\iota^{k}})+\frac{\partial}{\partial x^{k}}(L-\rho\frac{\partial^{r}L}{\partial\rho})=0$ (23)$\phi$ $\sqrt{}$) ( $\Delta\phi$ $\Delta\psi$ ) $I$:$\Delta\phi$ : $\partial_{t}\rho+\nabla\cdot(\rho v)=0$ ( ), (24)$\Delta\psi$ : $\partial_{t}(\rho s)+\nabla\cdot(\rho sv)=0$ . (25)53 ( )(23) , $m_{k}$ $M_{k}^{l}$:$m_{k}= \frac{\partial’L}{\partial’v^{k}}$ , $M_{k}^{l}=v^{l} \frac{\partial L}{\partial v^{k}}+(L-\rho\frac{\partial L}{\partial\rho})\delta_{k}^{l}$ . (26)(7) , $m_{k}=\rho v_{k}$ $\Lambda l_{k}^{l}=\rho v_{k^{V^{\downarrow}}}+p\delta_{k}^{l}$ ($v_{k}=v^{k})$ . (23) $\partial_{t}(\rho v^{k})+\partial_{l}’(\rho v^{l}v^{k})+\partial_{kp=}0(\partial_{k}=\partial/\partial x^{k})$. (24) :$\partial_{t}v^{k}+(v^{l}\partial_{l})v^{k}=-\frac{1}{\rho}’\partial_{k}p$ $(=-’\partial_{k}h)$ . (27)(16) . (14) :$j)_{t}[ \rho(\frac{1}{2}\uparrow^{2}+\epsilon)]+\partial_{k}[\rho?)^{k}(\frac{1}{2}\uparrow^{2}+$ $)]=0$ .6$SO(3)$so(3) $[12|$ $\nabla_{t}$ , $v$$A$ :$\nabla_{t}$ $=$ $\partial_{t}+(v\cdot\nabla)$ , (28)$v=\nabla_{t}x$ $=$ $(\partial_{t}+(v\cdot\nabla))x$ , (29)$\mathcal{A}=\nabla_{t}v$ $=$ $\partial_{t}v$ $(v\cdot\nabla)v$ (30)$\nabla_{t}v$ $=$ $\partial_{t}v+grad(\frac{1}{2}v^{2})+\omega xv$ . (31)$\nabla_{t}v=\partial_{t}v+\nabla(\frac{1}{2}v^{2})+\omega\cross v$$\omega\equiv\nabla\cross v$ $\nabla_{t}v$$(t, x, v)$ $U$ $(t_{*}, x_{*}, v_{*})$$t_{*}=t,$ $x_{*}=x-Ut,$ $v_{*}=v-U$: $\nabla_{t}v=(\nabla_{t}v)_{*}$ .227 LagrangianLagrangian . 51(20) 2 $\partial_{\tau}(\cdot)$ . $I= \int\int[\Lambda_{T}+\partial_{\tau}(\cdot)]d\tau d^{3}a$$[14]\sim[17|$ .helicity( Hopf invariant, [18]) non-trivial topologyChem-Simons term $I$ 4 1$\partial_{\mu}F^{\mu}$51 $\tau$ .7.1 Lagrangian $\Lambda_{A}$(11) $\tau$ . $(a, b, c)$curl$\nabla_{a}\cross\partial_{\tau}V_{a}=\partial_{\tau}(\nabla_{a}\cross V_{a})=0$ , (32). $\nabla_{a}=(\partial_{a}, \partial_{b}, \partial_{c})$ . $\nabla_{a}xV$ $=\Omega_{a}(a)$ .$V_{a}$ $x$- $v=(X_{\tau}, Y_{\tau}, Z_{\tau})=(u, v, w)$ $a$- . 1-$V^{1}$ :$V^{1}$$=$ $V_{a}da+V_{b}db+V_{c}$ $dc$ (33)$=$ $udx+vdy+wdz$ . (34)$V_{a}=ux_{a}+vy_{a}+uz_{a},$ $x_{a}=\partial X/\partial ra,$ $u=X_{\tau}$ , etc. . $dV^{1}$ 2- $\Omega^{2}=dV^{1}$ :$\Omega^{2}$$=$ $\Omega_{a}db\wedge dc+\Omega_{b}dc\wedge da+\Omega_{c}da\wedge db$$=$ $\omega_{T}dy\wedge dz+\omega_{y}dz\wedge dx+\omega_{z}dx\wedge dy$, (35)$(\Omega_{a}, \Omega_{b}, \Omega_{c})=\Omega_{a}$ . $\nabla xv=(\omega_{x}, \omega_{y}, \omega_{z})=\omega$ . $\Omega_{a}$ X-$\omega$ $a$- . (32) 2- $\Omega^{2}$ $\tau$-Lie : $\mathcal{L}_{\partial_{\tau}}\Omega^{2}=0$.$a$- $A_{a}=(\overline{\mathcal{A}}_{a}, \overline{A}_{b}, \overline{A}_{c})$ 1- $A^{1}$$\mathcal{A}^{1}=\overline{A}_{a}da+\overline{A}_{b}db+\overline{A}_{c}$dc $=\overline{A}_{x}dx+\overline{A}_{y}dy+\overline{A}_{z}d_{\overline{k}}$ . Lagrangian:$A$ 4 $=-/M\langle f\partial_{\tau}A_{a},$ $\Omega_{a}\rangle d^{3}a=/\Lambda f\langle A,$ $E_{W}[\omega|\rangle d^{3}x$ ,$E_{W}[\omega]\equiv\partial_{t}\omega+(v\cdot\nabla)\omega-(\omega\cdot\nabla)v+(\nabla\cdot v)\omega$.Lagrangian 3 [12] : (i) $v$ ,(ii) $A$ :$E_{W}[\omega|=\partial_{t}\omega+(v\cdot\nabla)\omega-(\omega\cdot\nabla)v+(\nabla\cdot v)\omega=0$,(iii) $H$ . ,$H=/V \omega\cdot vd^{3_{X}}=/|^{f}\omega\cdot\frac{E_{W}[cur1A|}{\rho}d^{3_{X}}$ ,$E_{W}[B|$ $(B=cur1A)$ .2372Lagrangiana Eulerian x $(a)$ $\partial x^{k}/\partial a^{l}$ 9. 53 6 $v=$$(X_{\tau}, Y_{\tau}, Z_{\tau})$ $(V_{a}, V_{b}, V_{c})$ 3 (12), $A=(X_{\tau \mathcal{T}^{1}}Y_{\tau\tau}, Z_{\tau\tau})$ $(A_{a}, A_{b}, A_{c})$3 (15) . 3 $\omega=(\omega_{x},\omega_{y}, \omega_{z})$ $\Omega_{a}(a)=$$(\Omega_{a}, \Omega_{b}, \Omega_{c})$ (35) . $\Omega_{a}$ :$\Omega_{a}$ $=$ $\omega_{x}(’\partial_{b}y\partial_{c}z-\partial_{c}y\partial_{b}z)+\omega_{y}(\partial_{b}z\partial_{c}x-\partial_{c}z\partial_{b}x)$$+\omega_{z}(\partial_{b^{X}}\partial_{c}y-\partial_{c}x\partial_{b}y)$ . (36)$x$ $a$ 2 3: . 3 9$\partial x^{k}/\partial a^{l}$ . , Lagrangiana Eulerian x$(a)$[12].8 discussion.(35) 39 $\partial x^{k}/\partial x^{l}$ , Lagrangian a–Eulerianx$(a)$-[1] C. Truesdell, 1960, “Rational Mechanics, 1687-1788”, Arch. Hist. of Exac. Sci. 1, 3-36.[2] C. Ikuaedell, 1954, “Rational Fluid Mechanics, 1687-1765”, LEOO, .II, Vol.12, &125.[3] , , , 1997.$[4|$ , 250 .[5] L. Euler, 1736, $[E15,E16|$ “Mechanica, sive motus scientia analytice exposita” (Mechanics,motion of science of analytic exposition), LEOO, Ser.II, Vol.1, 2.[6] L. Euler, 1750,[El77] “ Decouverte $d$ ’ un nouveau principe de Mecanique” (Discovery of a newprinciple of mochanics), LEOO, Ser.II, Vol.5.[7] L. Euler, 1752, [E258] “ Principia motus fluidorum” (Principles of the motions of fluids), LEOO,Ser.II, Vol.12, 133-168.[8] L. Euler, 1753, [E225] “ Principes generaux de l’etat d’ equilibre des fluides” (General Principlesof the state of equilibrium of fluids), LEOO Ser.II, vol.12, 2-53. (Berlin 1757).[9] L. Euler, 1755, $[E226|$ “Principes gcneraux du mouvement des fluides” (Gencral Principles ofthe motions of fluids), LEOO, Ser.II, vol.12, 54-91. (Berlin 1757). : ”General laws of themotion of fluids”, Fluid Dynamics, vol.34, no.6, 801-822 (1999).24[10] T. Kambe, 2007, “Variational formulation of the motion of an ideal fluid on the basisof gauge principle“, Preprint: htti $://a\zeta_{t}Y$i.v $ewg’jak_{\lrcorner}\cdot s1t\}_{\ell}^{r}f7.177^{r}d$, accpted to Physica $D$ (2008),Proc. of the Conference ”Euler Equations: 250 Years On“ (2007, France).[11] T. Kambe 2007, “Gauge $pri_{I1Cl\dot{\beta}}$le and variationaJ formulation for idealfluidswith reference to translation $symmetr \oint’$ , Fluid Dyn. Res. 39, 98–120.[12] T. Kambe 2007, “Variational formulation of ideal fluid flows according to gauge principle”,accepted to Fluid Dyn. Res; Preprint $TK2007b$ , htt$tJ:l1$www. urole.dti.ne.$i\ddagger$)$/kambe/$[13] H.Lamb, Hydrodynamics. Cambridge University Press, Cambridge, 1932.[14] S.S. Chern, Complex manifolds withoutpotential theory 2nd ed., Springer-Verlag,Berlin, 1979.[15] R. Jackiw, ‘Chern-Simons terms and coeycles in physies and mathematies”, in FradkinFestschrift, Adam Hilger, Bristol, 1985.[16] $\}$ S. Desser, R. Jackiw $R$ and S. Templeton, Annals ofPhysidi, 140 (1982) 372 $\cdot 411$ .[181 R. Jaekiw, Lectures on FluidMechanics (Springer, 2002).[191 VI. Arnold and B. Khesin, Topologicalmethods in hydrodynamics, Springer,$lArnold998$1998.2007 [8] 250“Euler Equations: 250 Years On”. Aussois CNRS –( )6 7ProceedingsPhysica $D$ : Nonlinear Phenomena (2008).EE250 http:$//www$.obs-nice.&/etc7/EE250/.CNRS $($ $)$25

オイラー ノ ホウテイシキ ドウシュツ ト リュウタイ ウンドウ ノ ゲージ リロン オイラー ホウテイシキ 250ネン レンゾクタイ リキガク ニオケル オイラー ノ イサン

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オイラーノホウテイシキドウシュツトリュウタイウンドウノゲージリロンオイラーホウテイシキ 250ネンレンゾクタイリキガクニオケルオイラーノイサン

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オイラー ノ ホウテイシキ ドウシュツ ト リュウタイ ウンドウ ノ ゲージ リロン オイラー ホウテイシキ 250ネン レンゾクタイ リキガク ニオケル オイラー ノ イサン

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オイラーノホウテイシキドウシュツトリュウタイウンドウノゲージリロンオイラーホウテイシキ 250ネンレンゾクタイリキガクニオケルオイラーノイサン